Estimation of Main Effect When Covariates Have Non-Proportional Hazards

The Cox proportional hazards (PH) regression model has been widely used to analyze survival data in clinical trials and observational studies. In addition to estimating the main treatment or exposure group effect, it is common to adjust for additional covariates using the Cox model. It is well known that violation of the PH assumption can lead to estimates that are biased and difficult to interpret, and model checking has become a routine procedure. However, such checking might focus on the primary group comparisons, and the assumption can still be violated when adjusting for many of the potential covariates. We study the effect of violation of the PH assumption of the covariates on the estimation of the main group effect in the Cox model. The results are summarized in terms of the bias and the coverage properties of the confidence intervals. Overall in randomized clinical trials, the bias caused by misspecifying the PH assumption on the covariates is no more than 15% in absolute value regardless of sample size. In observational studies where the covariates are likely correlated with the group variable, however, the bias can be very severe. The coverage properties largely depend on sample size, as expected, as bias becomes dominating with increasing sample size. These findings should serve as cautionary notes when adjusting for potential confounders in observational studies, as the violation of PH assumption on the confounders can lead to erroneous results.     

Estimation of average causal effect using the restricted mean residual lifetime as effect measure

Although mean residual lifetime is often of interest in biomedical studies, restricted mean residual lifetime must be considered in order to accommodate censoring. Differences in the restricted mean residual lifetime can be used as an appropriate quantity for comparing different treatment groups with respect to their survival times. In observational studies where the factor of interest is not randomized, covariate adjustment is needed to take into account imbalances in confounding factors. In this article, we develop an estimator for the average causal treatment difference using the restricted mean residual lifetime as target parameter. We account for confounding factors using the Aalen additive hazards model. Large sample property of the proposed estimator is established and simulation studies are conducted in order to assess small sample performance of the resulting estimator. The method is also applied to an observational data set of patients after an acute myocardial infarction event.     

Sample size determination for clustered right-censored data using copula models

Determination of an adequate sample size is critical to the design of research ventures. For clustered right-censored data, Manatunga and Chen [Sample size estimation for survival outcomes in cluster-randomized studies with small cluster sizes. Biometrics. 2000;56(2):616–621] proposed a sample size calculation based on considering the bivariate marginal distribution as Clayton copula model. In addition to the Clayton copula, other important family of copulas, such as Gumbel and Frank copulas are also well established in multivariate survival analysis. However, sample size calculation based on these assumptions has not been fully investigated yet. To broaden the scope of Manatunga and Chen [Sample size estimation for survival outcomes in cluster-randomized studies with small cluster sizes. Biometrics. 2000;56(2):616–621]'s research and achieve a more flexible sample size calculation for clustered right-censored data, we extended the work by assuming the marginal distribution as bivariate Gumbel and Frank copulas. We evaluate the performance of the proposed method and investigate the impacts of the accrual times, follow-up times and the within-clustered correlation effect of the study. The proposed method is applied to two real-world studies, and the R code is made available to users.     

Calculating power for the comparison of dependent κ-coefficients

Summary. In the psychosocial and medical sciences, some studies are designed to assess the agreement between different raters and/or different instruments. Often the same sample will be used to compare the agreement between two or more assessment methods for simplicity and to take advantage of the positive correlation of the ratings. Although sample size calculations have become an important element in the design of research projects, such methods for agreement studies are scarce. We adapt the generalized estimating equations approach for modelling dependent κ -statistics to estimate the sample size that is required for dependent agreement studies. We calculate the power based on a Wald test for the equality of two dependent κ -statistics. The Wald test statistic has a non-central χ ²-distribution with non-centrality parameter that can be estimated with minimal assumptions. The method proposed is useful for agreement studies with two raters and two instruments, and is easily extendable to multiple raters and multiple instruments. Furthermore, the method proposed allows for rater bias. Power calculations for binary ratings under various scenarios are presented. Analyses of two biomedical studies are used for illustration.     

Sample Size Calculation for Rank Tests Comparing K Survival Distributions

Rank tests, such as logrank or Wilcoxon rank sum tests, have been popularly used to compare survival distributions of two or more groups in the presence of right censoring. However, there has been little research on sample size calculation methods for rank tests to compare more than two groups. An existing method is based on a crude approximation, which tends to underestimate sample size, i.e., the calculated sample size has lower power than projected. In this paper we propose an asymptotically correct method and an approximate method for sample size calculation. The proposed methods are compared to other methods through simulation studies.     

Monte Carlo Approximations of the Quantiles of a Sample Statistic

We consider in this article the problem of numerically approximating the quantiles of a sample statistic for a given population, a problem of interest in many applications, such as bootstrap confidence intervals. The proposed Monte Carlo method can be routinely applied to handle complex problems that lack analytical results. Furthermore, the method yields estimates of the quantiles of a sample statistic of any sample size though Monte Carlo simulations for only two optimally selected sample sizes are needed. An analysis of the Monte Carlo design is performed to obtain the optimal choices of these two sample sizes and the number of simulated samples required for each sample size. Theoretical results are presented for the bias and variance of the numerical method proposed. The results developed are illustrated via simulation studies for the classical problem of estimating a bivariate linear structural relationship. It is seen that the size of the simulated samples used in the Monte Carlo method does not have to be very large and the method provides a better approximation to quantiles than those based on an asymptotic normal theory for skewed sampling distributions.     

Sample size implications when biases are modelled rather than ignored

Summary.  Realistic statistical modelling of observational data often suggests a statistical model which is not fully identified, owing to potential biases that are not under the control of study investigators. Bayesian inference can be implemented with such a model, ideally with the most precise prior knowledge that can be ascertained. However, as a consequence of the non-identifiability, inference cannot be made arbitrarily accurate by choosing the sample size to be sufficiently large. In turn, this has consequences for sample size determination. The paper presents a sample size criterion that is based on a quantification of how much Bayesian learning can arise in a given non-identified model. A global perspective is adopted, whereby choosing larger sample sizes for some studies necessarily implies that some other potentially worthwhile studies cannot be undertaken. This suggests that smaller sample sizes should be selected with non-identified models, as larger sample sizes constitute a squandering of resources in making estimator variances very small compared with their biases. Particularly, consider two investigators planning the same study, one of whom admits to the potential biases at hand and consequently uses a non-identified model, whereas the other pretends that there are no biases, leading to an identified but less realistic model. It is seen that the former investigator always selects a smaller sample size than the latter, with the difference being quite marked in some illustrative cases.     

Inferences from logistic regression models in the presence of small samples,rare events,nonlinearity, and multicollinearity with observational data

The logistic regression model has been widely used in the social and natural sciences and results from studies using this model can have significant policy impacts. Thus, confidence in the reliability of inferences drawn from these models is essential. The robustness of such inferences is dependent on sample size. The purpose of this article is to examine the impact of alternative data sets on the mean estimated bias and efficiency of parameter estimation and inference for the logistic regression model with observational data. A number of simulations are conducted examining the impact of sample size, nonlinear predictors, and multicollinearity on substantive inferences (e.g. odds ratios, marginal effects) when using logistic regression models. Findings suggest that small sample size can negatively affect the quality of parameter estimates and inferences in the presence of rare events, multicollinearity, and nonlinear predictor functions, but marginal effects estimates are relatively more robust to sample size.     

Efficient sample size allocation with cost constraints for heterogeneous-variance group comparison

When conducting research with controlled experiments, sample size planning is one of the important decisions that researchers have to make. However, current methods do not adequately address this issue with regard to variance heterogeneity with some cost constraints for comparing several treatment means. This paper proposes a sample size allocation ratio in the fixed-effect heterogeneous analysis of variance when group variances are unequal and in cases where the sampling and/or variable cost has some constraints. The efficient sample size allocation is determined for the purpose of minimizing total cost with a designated power or maximizing the power with a given total cost. Finally, the proposed method is verified by using the index of relative efficiency and the corresponding total cost and the total sample size needed. We also apply our method in a pain management trial to decide an efficient sample size. Simulation studies also show that the proposed sample size formulas are efficient in terms of statistical power. SAS and R codes are provided in the appendix for easy application.     

Design issues for studies of infectious diseases

The design of infectious disease studies has received little attention because they are generally viewed as observational studies. That is, epidemic and endemic disease transmission happens and we observe it. We argue here that statistical design often provides useful guidance for such studies with regard to type of data and the size of the data set to be collected. It is shown that data on disease transmission in part of the community enables the estimation of central parameters and it is possible to compute the sample size required to make inferences with a desired precision. We illustrate this for data on disease transmission in a single community of uniformly mixing individuals and for data on outbreak sizes in households. Data on disease transmission is usually incomplete and this creates an identifiability problem for certain parameters of multitype epidemic models. We identify designs that can overcome this problem for the important objective of estimating parameters that help to assess the effectiveness of a vaccine. With disease transmission in animal groups there is greater scope for conducting planned experiments and we explore some possibilities for such experiments. The topic is largely unexplored and numerous open research problems in the area of statistical design of infectious disease data are mentioned.     

Sample size calculation for a proportional hazards mixture cure model with nonbinary covariates

Sample size calculation is a critical issue in clinical trials because a small sample size leads to a biased inference and a large sample size increases the cost. With the development of advanced medical technology, some patients can be cured of certain chronic diseases, and the proportional hazards mixture cure model has been developed to handle survival data with potential cure information. Given the needs of survival trials with potential cure proportions, a corresponding sample size formula based on the log-rank test statistic for binary covariates has been proposed by Wang et al. [25]. However, a sample size formula based on continuous variables has not been developed. Herein, we presented sample size and power calculations for the mixture cure model with continuous variables based on the log-rank method and further modified it by Ewell's method. The proposed approaches were evaluated using simulation studies for synthetic data from exponential and Weibull distributions. A program for calculating necessary sample size for continuous covariates in a mixture cure model was implemented in R.     

Events per variable for risk differences and relative risks using pseudo-observations

A method based on pseudo-observations has been proposed for direct regression modeling of functionals of interest with right-censored data, including the survival function, the restricted mean and the cumulative incidence function in competing risks. The models, once the pseudo-observations have been computed, can be fitted using standard generalized estimating equation software. Regression models can however yield problematic results if the number of covariates is large in relation to the number of events observed. Guidelines of events per variable are often used in practice. These rules of thumb for the number of events per variable have primarily been established based on simulation studies for the logistic regression model and Cox regression model. In this paper we conduct a simulation study to examine the small sample behavior of the pseudo-observation method to estimate risk differences and relative risks for right-censored data. We investigate how coverage probabilities and relative bias of the pseudo-observation estimator interact with sample size, number of variables and average number of events per variable.     

Approximate MLEs of the parameters of location-scale models under type II censoring

Log-location-scale distributions are widely used parametric models that have fundamental importance in both parametric and semiparametric frameworks. The likelihood equations based on a Type II censored sample from location-scale distributions do not provide explicit solutions for the para-meters. Statistical software is widely available and is based on iterative methods (such as, Newton Raphson Algorithm, EM algorithm etc.), which require starting values near the global maximum. There are also many situations that the specialized software does not handle. This paper provides a method for determining explicit estimators for the location and scale parameters by approximating the likelihood function, where the method does not require any starting values. The performance of the proposed approximate method for the Weibull distribution and Log-Logistic distributions is compared with those based on iterative methods through the use of simulation studies for a wide range of sample size and Type II censoring schemes. Here we also examine the probability coverages of the pivotal quantities based on asymptotic normality. In addition, two examples are given.     

Empirical Likelihood Based Rank Regression Inference

Rank regression procedures have been proposed and studied for numerous research applications that do not satisfy the underlying assumptions of the more common linear regression models. This article develops confidence regions for the slope parameter of rank regression using an empirical likelihood (EL) ratio method. It has the advantage of not requiring variance estimation which is required for the normal approximation method. The EL method is also range respecting and results in asymmetric confidence intervals. Simulation studies are used to compare and evaluate normal approximation versus EL inference methods for various conditions such as different sample size or error distribution. The simulation study demonstrates our proposed EL method almost outperforms the traditional method in terms of coverage probability, lower-tail side error, and upper-tail side error. An application of stability analysis also shows the EL method results in shorter confidence intervals for real life data.     

A semiparametric pseudo-score method for analysis of two-phase studies with continuous phase-I covariates

Two-phase study designs can reduce cost and other practical burdens associated with large scale epidemiologic studies by limiting ascertainment of expensive covariates to a smaller but informative sub-sample (phase-II) of the main study (phase-I). During the analysis of such studies, however, subjects who are selected at phase-I but not at phase-II, remain informative as they may have partial covariate information. A variety of semi-parametric methods now exist for incorporating such data from phase-I subjects when the covariate information can be summarized into a finite number of strata. In this article, we consider extending the pseudo-score approach proposed by Chatterjee et al. (J Am Stat Assoc 98:158–168, 2003) using a kernel smoothing approach to incorporate information on continuous phase-I covariates. Practical issues and algorithms for implementing the methods using existing software are discussed. A sandwich-type variance estimator based on the influence function representation of the pseudo-score function is proposed. Finite sample performance of the methods are studies using simulated data. Advantage of the proposed smoothing approach over alternative methods that use discretized phase-I covariate information is illustrated using two-phase data simulated within the National Wilms Tumor Study (NWTS).     

Robust feature screening for high-dimensional survival data

Ultra-high dimensional data arise in many fields of modern science, such as medical science, economics, genomics and imaging processing, and pose unprecedented challenge for statistical analysis. With such rapid-growth size of scientific data in various disciplines, feature screening becomes a primary step to reduce the high dimensionality to a moderate scale that can be handled by the existing penalized methods. In this paper, we introduce a simple and robust feature screening method without any model assumption to tackle high dimensional censored data. The proposed method is model-free and hence applicable to a general class of survival models. The sure screening and ranking consistency properties without any finite moment condition of the predictors and the response are established. The computation of the proposed method is rather straightforward. Finite sample performance of the newly proposed method is examined via extensive simulation studies. An application is illustrated with the gene association study of the mantle cell lymphoma.     

A Bayesian sequential design with adaptive randomization for 2‐sided hypothesis test

Bayesian sequential and adaptive randomization designs are gaining popularity in clinical trials thanks to their potentials to reduce the number of required participants and save resources. We propose a Bayesian sequential design with adaptive randomization rates so as to more efficiently attribute newly recruited patients to different treatment arms. In this paper, we consider 2‐arm clinical trials. Patients are allocated to the 2 arms with a randomization rate to achieve minimum variance for the test statistic. Algorithms are presented to calculate the optimal randomization rate, critical values, and power for the proposed design. Sensitivity analysis is implemented to check the influence on design by changing the prior distributions. Simulation studies are applied to compare the proposed method and traditional methods in terms of power and actual sample sizes. Simulations show that, when total sample size is fixed, the proposed design can obtain greater power and/or cost smaller actual sample size than the traditional Bayesian sequential design. Finally, we apply the proposed method to a real data set and compare the results with the Bayesian sequential design without adaptive randomization in terms of sample sizes. The proposed method can further reduce required sample size.     

Sample size calculation for an agreement study

It is often necessary to compare two measurement methods in medicine and other experimental sciences. This problem covers a broad range of data. Many authors have explored ways of assessing the agreement of two sets of measurements. However, there has been relatively little attention to the problem of determining sample size for designing an agreement study. In this paper, a method using the interval approach for concordance is proposed to calculate sample size in conducting an agreement study. The philosophy behind this is that the concordance is satisfied when no more than the pre‐specified k discordances are found for a reasonable large sample size n since it is much easier to define a discordance pair. The goal here is to find such a reasonable large sample size n. The sample size calculation is based on two rates: the discordance rate and tolerance probability, which in turn can be used to quantify an agreement study. The proposed approach is demonstrated through a real data set. Copyright © 2009 John Wiley & Sons, Ltd.     

A simple test procedure in standardizing the power of Hosmer–Lemeshow test in large data sets

The Hosmer–Lemeshow (H–L) test is a widely used method when assessing the goodness-of-fit of a logistic regression model. However, the H–L test is sensitive to the sample sizes and the number of groups in H–L test. Cautions need to be taken for interpreting an H–L test with a large sample size. In this paper, we propose a simple test procedure to evaluate the model fit of logistic regression model with a large sample size, in which a bootstrap method is used and the test result is determined by the power of H–L test at the target sample size. Simulation studies show that the proposed method can effectively standardize the power of the H–L test under the pre-specified level of type I error. Application to the two datasets illustrates the usefulness of the proposed model.     

Sequential design approaches for bioequivalence studies with crossover designs

The planning of bioequivalence (BE) studies, as for any clinical trial, requires a priori specification of an effect size for the determination of power and an assumption about the variance. The specified effect size may be overly optimistic, leading to an underpowered study. The assumed variance can be either too small or too large, leading, respectively, to studies that are underpowered or overly large. There has been much work in the clinical trials field on various types of sequential designs that include sample size reestimation after the trial is started, but these have seen only little use in BE studies. The purpose of this work was to validate at least one such method for crossover design BE studies. Specifically, we considered sample size reestimation for a two-stage trial based on the variance estimated from the first stage. We identified two methods based on Pocock's method for group sequential trials that met our requirement for at most negligible increase in type I error rate.